Additive manufacturing techniques which enable control over the placement and orientation of particles within composite inks can produce structures with tailored gradients in structural and functional properties. One such technique is direct ink writing with acoustophoresis (DIWA), wherein a composite ink is extruded through a direct-write nozzle containing a standing bulk acoustic wave which aligns and positions particles. Driving force-based scaling relationships contextualize processing-structure relationships in DIWA. In a series of experiments which progress in geometric complexity from basic primitives to complete structures, a physical framework is constructed for controlling filament microstructures and external geometries in DIWA. In isolated filaments, there are trade-offs between focusing and form holding. Increasing the ink viscosity, increasing the print speed, and decreasing the acoustic wave amplitude widen the spatial distribution of particles in agreement with scaling relationships for acoustophoresis, but more viscous inks improve form holding. In the print bead between the nozzle and substrate, digital image analysis is used to measure filament stability, nozzle wetting, and rotational flows in the low-viscosity inks required for acoustophoresis. Viscocapillary lubrication theory accurately predicts the bounds of stability, and the contact line position and angle can be used to detect the beginnings of filament rupture, allowing for algorithms which prevent rupture in-situ. In polygonal prisms, the internal structure of filaments changes during deposition into layer-by-layer and bath support gels. Filament microstructures change during deposition, during relaxation, and when the nozzle returns to write neighboring lines. Experimental flow fields and particle distributions suggest that inertia and viscoplasticity influence the filament microstructure just after deposition and the microstructure of neighboring filaments, and interfacial energy and gravity cause filaments to spread after deposition. An analytical model is proposed to diagnose sources of direction dependent microstructures as a function of acoustics, inertia, viscous dissipation, and stage calibration. The support geometry can be used to accentuate or suppress aspects of this direction dependence. Finally, inertia swells written corners, and capillarity smooths written corners, leading to distortions in filament microstructures at corners. Bath support suppresses these corner defects.